| I am currently examining the global behavior of solutions to semilinear parabolic equations with certain polynomial nonlinearities. My work lies at the intersection of several mathematical topics. On one hand are traditional topics in partial differential equations, such as maximum principles and numerical approximation. On the other are newer topics from bifurcations of dynamical systems and the topology of function spaces. I build intuition in one of these fields, and then employ methods from the others for proofs. The analogy of parabolic equations with systems of ordinary differential equations has been particularly fruitful, especially where it falls apart. For instance, a critical point of a parabolic equation whose linearization is stable need not be stable itself. In this case, understanding the topology of the (infinite dimensional) stable and unstable manifolds for the critical point is essential, and provides much of the motivation for my work. |
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